Step |
Hyp |
Ref |
Expression |
1 |
|
fiuncmp.1 |
⊢ 𝑋 = ∪ 𝐽 |
2 |
|
ssid |
⊢ 𝐴 ⊆ 𝐴 |
3 |
|
simp2 |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) → 𝐴 ∈ Fin ) |
4 |
|
sseq1 |
⊢ ( 𝑡 = ∅ → ( 𝑡 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴 ) ) |
5 |
|
iuneq1 |
⊢ ( 𝑡 = ∅ → ∪ 𝑥 ∈ 𝑡 𝐵 = ∪ 𝑥 ∈ ∅ 𝐵 ) |
6 |
|
0iun |
⊢ ∪ 𝑥 ∈ ∅ 𝐵 = ∅ |
7 |
5 6
|
eqtrdi |
⊢ ( 𝑡 = ∅ → ∪ 𝑥 ∈ 𝑡 𝐵 = ∅ ) |
8 |
7
|
oveq2d |
⊢ ( 𝑡 = ∅ → ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑡 𝐵 ) = ( 𝐽 ↾t ∅ ) ) |
9 |
8
|
eleq1d |
⊢ ( 𝑡 = ∅ → ( ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑡 𝐵 ) ∈ Comp ↔ ( 𝐽 ↾t ∅ ) ∈ Comp ) ) |
10 |
4 9
|
imbi12d |
⊢ ( 𝑡 = ∅ → ( ( 𝑡 ⊆ 𝐴 → ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑡 𝐵 ) ∈ Comp ) ↔ ( ∅ ⊆ 𝐴 → ( 𝐽 ↾t ∅ ) ∈ Comp ) ) ) |
11 |
10
|
imbi2d |
⊢ ( 𝑡 = ∅ → ( ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) → ( 𝑡 ⊆ 𝐴 → ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑡 𝐵 ) ∈ Comp ) ) ↔ ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) → ( ∅ ⊆ 𝐴 → ( 𝐽 ↾t ∅ ) ∈ Comp ) ) ) ) |
12 |
|
sseq1 |
⊢ ( 𝑡 = 𝑦 → ( 𝑡 ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴 ) ) |
13 |
|
iuneq1 |
⊢ ( 𝑡 = 𝑦 → ∪ 𝑥 ∈ 𝑡 𝐵 = ∪ 𝑥 ∈ 𝑦 𝐵 ) |
14 |
13
|
oveq2d |
⊢ ( 𝑡 = 𝑦 → ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑡 𝐵 ) = ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑦 𝐵 ) ) |
15 |
14
|
eleq1d |
⊢ ( 𝑡 = 𝑦 → ( ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑡 𝐵 ) ∈ Comp ↔ ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑦 𝐵 ) ∈ Comp ) ) |
16 |
12 15
|
imbi12d |
⊢ ( 𝑡 = 𝑦 → ( ( 𝑡 ⊆ 𝐴 → ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑡 𝐵 ) ∈ Comp ) ↔ ( 𝑦 ⊆ 𝐴 → ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑦 𝐵 ) ∈ Comp ) ) ) |
17 |
16
|
imbi2d |
⊢ ( 𝑡 = 𝑦 → ( ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) → ( 𝑡 ⊆ 𝐴 → ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑡 𝐵 ) ∈ Comp ) ) ↔ ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) → ( 𝑦 ⊆ 𝐴 → ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑦 𝐵 ) ∈ Comp ) ) ) ) |
18 |
|
sseq1 |
⊢ ( 𝑡 = ( 𝑦 ∪ { 𝑧 } ) → ( 𝑡 ⊆ 𝐴 ↔ ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐴 ) ) |
19 |
|
iuneq1 |
⊢ ( 𝑡 = ( 𝑦 ∪ { 𝑧 } ) → ∪ 𝑥 ∈ 𝑡 𝐵 = ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ) |
20 |
19
|
oveq2d |
⊢ ( 𝑡 = ( 𝑦 ∪ { 𝑧 } ) → ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑡 𝐵 ) = ( 𝐽 ↾t ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ) ) |
21 |
20
|
eleq1d |
⊢ ( 𝑡 = ( 𝑦 ∪ { 𝑧 } ) → ( ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑡 𝐵 ) ∈ Comp ↔ ( 𝐽 ↾t ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ) ∈ Comp ) ) |
22 |
18 21
|
imbi12d |
⊢ ( 𝑡 = ( 𝑦 ∪ { 𝑧 } ) → ( ( 𝑡 ⊆ 𝐴 → ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑡 𝐵 ) ∈ Comp ) ↔ ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐴 → ( 𝐽 ↾t ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ) ∈ Comp ) ) ) |
23 |
22
|
imbi2d |
⊢ ( 𝑡 = ( 𝑦 ∪ { 𝑧 } ) → ( ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) → ( 𝑡 ⊆ 𝐴 → ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑡 𝐵 ) ∈ Comp ) ) ↔ ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) → ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐴 → ( 𝐽 ↾t ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ) ∈ Comp ) ) ) ) |
24 |
|
sseq1 |
⊢ ( 𝑡 = 𝐴 → ( 𝑡 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴 ) ) |
25 |
|
iuneq1 |
⊢ ( 𝑡 = 𝐴 → ∪ 𝑥 ∈ 𝑡 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐵 ) |
26 |
25
|
oveq2d |
⊢ ( 𝑡 = 𝐴 → ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑡 𝐵 ) = ( 𝐽 ↾t ∪ 𝑥 ∈ 𝐴 𝐵 ) ) |
27 |
26
|
eleq1d |
⊢ ( 𝑡 = 𝐴 → ( ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑡 𝐵 ) ∈ Comp ↔ ( 𝐽 ↾t ∪ 𝑥 ∈ 𝐴 𝐵 ) ∈ Comp ) ) |
28 |
24 27
|
imbi12d |
⊢ ( 𝑡 = 𝐴 → ( ( 𝑡 ⊆ 𝐴 → ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑡 𝐵 ) ∈ Comp ) ↔ ( 𝐴 ⊆ 𝐴 → ( 𝐽 ↾t ∪ 𝑥 ∈ 𝐴 𝐵 ) ∈ Comp ) ) ) |
29 |
28
|
imbi2d |
⊢ ( 𝑡 = 𝐴 → ( ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) → ( 𝑡 ⊆ 𝐴 → ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑡 𝐵 ) ∈ Comp ) ) ↔ ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) → ( 𝐴 ⊆ 𝐴 → ( 𝐽 ↾t ∪ 𝑥 ∈ 𝐴 𝐵 ) ∈ Comp ) ) ) ) |
30 |
|
rest0 |
⊢ ( 𝐽 ∈ Top → ( 𝐽 ↾t ∅ ) = { ∅ } ) |
31 |
|
0cmp |
⊢ { ∅ } ∈ Comp |
32 |
30 31
|
eqeltrdi |
⊢ ( 𝐽 ∈ Top → ( 𝐽 ↾t ∅ ) ∈ Comp ) |
33 |
32
|
3ad2ant1 |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) → ( 𝐽 ↾t ∅ ) ∈ Comp ) |
34 |
33
|
a1d |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) → ( ∅ ⊆ 𝐴 → ( 𝐽 ↾t ∅ ) ∈ Comp ) ) |
35 |
|
ssun1 |
⊢ 𝑦 ⊆ ( 𝑦 ∪ { 𝑧 } ) |
36 |
|
id |
⊢ ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐴 → ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐴 ) |
37 |
35 36
|
sstrid |
⊢ ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐴 → 𝑦 ⊆ 𝐴 ) |
38 |
37
|
imim1i |
⊢ ( ( 𝑦 ⊆ 𝐴 → ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑦 𝐵 ) ∈ Comp ) → ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐴 → ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑦 𝐵 ) ∈ Comp ) ) |
39 |
|
simpl1 |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) ∧ ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐴 ∧ ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑦 𝐵 ) ∈ Comp ) ) → 𝐽 ∈ Top ) |
40 |
|
iunxun |
⊢ ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 = ( ∪ 𝑥 ∈ 𝑦 𝐵 ∪ ∪ 𝑥 ∈ { 𝑧 } 𝐵 ) |
41 |
|
simprr |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) ∧ ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐴 ∧ ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑦 𝐵 ) ∈ Comp ) ) → ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑦 𝐵 ) ∈ Comp ) |
42 |
|
cmptop |
⊢ ( ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑦 𝐵 ) ∈ Comp → ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑦 𝐵 ) ∈ Top ) |
43 |
|
restrcl |
⊢ ( ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑦 𝐵 ) ∈ Top → ( 𝐽 ∈ V ∧ ∪ 𝑥 ∈ 𝑦 𝐵 ∈ V ) ) |
44 |
43
|
simprd |
⊢ ( ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑦 𝐵 ) ∈ Top → ∪ 𝑥 ∈ 𝑦 𝐵 ∈ V ) |
45 |
41 42 44
|
3syl |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) ∧ ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐴 ∧ ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑦 𝐵 ) ∈ Comp ) ) → ∪ 𝑥 ∈ 𝑦 𝐵 ∈ V ) |
46 |
|
nfcv |
⊢ Ⅎ 𝑡 𝐵 |
47 |
|
nfcsb1v |
⊢ Ⅎ 𝑥 ⦋ 𝑡 / 𝑥 ⦌ 𝐵 |
48 |
|
csbeq1a |
⊢ ( 𝑥 = 𝑡 → 𝐵 = ⦋ 𝑡 / 𝑥 ⦌ 𝐵 ) |
49 |
46 47 48
|
cbviun |
⊢ ∪ 𝑥 ∈ { 𝑧 } 𝐵 = ∪ 𝑡 ∈ { 𝑧 } ⦋ 𝑡 / 𝑥 ⦌ 𝐵 |
50 |
|
vex |
⊢ 𝑧 ∈ V |
51 |
|
csbeq1 |
⊢ ( 𝑡 = 𝑧 → ⦋ 𝑡 / 𝑥 ⦌ 𝐵 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) |
52 |
50 51
|
iunxsn |
⊢ ∪ 𝑡 ∈ { 𝑧 } ⦋ 𝑡 / 𝑥 ⦌ 𝐵 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 |
53 |
49 52
|
eqtri |
⊢ ∪ 𝑥 ∈ { 𝑧 } 𝐵 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 |
54 |
51
|
oveq2d |
⊢ ( 𝑡 = 𝑧 → ( 𝐽 ↾t ⦋ 𝑡 / 𝑥 ⦌ 𝐵 ) = ( 𝐽 ↾t ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ) |
55 |
54
|
eleq1d |
⊢ ( 𝑡 = 𝑧 → ( ( 𝐽 ↾t ⦋ 𝑡 / 𝑥 ⦌ 𝐵 ) ∈ Comp ↔ ( 𝐽 ↾t ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ∈ Comp ) ) |
56 |
|
simpl3 |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) ∧ ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐴 ∧ ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑦 𝐵 ) ∈ Comp ) ) → ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) |
57 |
|
nfv |
⊢ Ⅎ 𝑡 ( 𝐽 ↾t 𝐵 ) ∈ Comp |
58 |
|
nfcv |
⊢ Ⅎ 𝑥 𝐽 |
59 |
|
nfcv |
⊢ Ⅎ 𝑥 ↾t |
60 |
58 59 47
|
nfov |
⊢ Ⅎ 𝑥 ( 𝐽 ↾t ⦋ 𝑡 / 𝑥 ⦌ 𝐵 ) |
61 |
60
|
nfel1 |
⊢ Ⅎ 𝑥 ( 𝐽 ↾t ⦋ 𝑡 / 𝑥 ⦌ 𝐵 ) ∈ Comp |
62 |
48
|
oveq2d |
⊢ ( 𝑥 = 𝑡 → ( 𝐽 ↾t 𝐵 ) = ( 𝐽 ↾t ⦋ 𝑡 / 𝑥 ⦌ 𝐵 ) ) |
63 |
62
|
eleq1d |
⊢ ( 𝑥 = 𝑡 → ( ( 𝐽 ↾t 𝐵 ) ∈ Comp ↔ ( 𝐽 ↾t ⦋ 𝑡 / 𝑥 ⦌ 𝐵 ) ∈ Comp ) ) |
64 |
57 61 63
|
cbvralw |
⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ↔ ∀ 𝑡 ∈ 𝐴 ( 𝐽 ↾t ⦋ 𝑡 / 𝑥 ⦌ 𝐵 ) ∈ Comp ) |
65 |
56 64
|
sylib |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) ∧ ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐴 ∧ ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑦 𝐵 ) ∈ Comp ) ) → ∀ 𝑡 ∈ 𝐴 ( 𝐽 ↾t ⦋ 𝑡 / 𝑥 ⦌ 𝐵 ) ∈ Comp ) |
66 |
|
ssun2 |
⊢ { 𝑧 } ⊆ ( 𝑦 ∪ { 𝑧 } ) |
67 |
|
simprl |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) ∧ ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐴 ∧ ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑦 𝐵 ) ∈ Comp ) ) → ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐴 ) |
68 |
66 67
|
sstrid |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) ∧ ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐴 ∧ ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑦 𝐵 ) ∈ Comp ) ) → { 𝑧 } ⊆ 𝐴 ) |
69 |
50
|
snss |
⊢ ( 𝑧 ∈ 𝐴 ↔ { 𝑧 } ⊆ 𝐴 ) |
70 |
68 69
|
sylibr |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) ∧ ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐴 ∧ ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑦 𝐵 ) ∈ Comp ) ) → 𝑧 ∈ 𝐴 ) |
71 |
55 65 70
|
rspcdva |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) ∧ ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐴 ∧ ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑦 𝐵 ) ∈ Comp ) ) → ( 𝐽 ↾t ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ∈ Comp ) |
72 |
|
cmptop |
⊢ ( ( 𝐽 ↾t ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ∈ Comp → ( 𝐽 ↾t ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ∈ Top ) |
73 |
|
restrcl |
⊢ ( ( 𝐽 ↾t ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ∈ Top → ( 𝐽 ∈ V ∧ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∈ V ) ) |
74 |
73
|
simprd |
⊢ ( ( 𝐽 ↾t ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ∈ Top → ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∈ V ) |
75 |
71 72 74
|
3syl |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) ∧ ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐴 ∧ ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑦 𝐵 ) ∈ Comp ) ) → ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∈ V ) |
76 |
53 75
|
eqeltrid |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) ∧ ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐴 ∧ ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑦 𝐵 ) ∈ Comp ) ) → ∪ 𝑥 ∈ { 𝑧 } 𝐵 ∈ V ) |
77 |
|
unexg |
⊢ ( ( ∪ 𝑥 ∈ 𝑦 𝐵 ∈ V ∧ ∪ 𝑥 ∈ { 𝑧 } 𝐵 ∈ V ) → ( ∪ 𝑥 ∈ 𝑦 𝐵 ∪ ∪ 𝑥 ∈ { 𝑧 } 𝐵 ) ∈ V ) |
78 |
45 76 77
|
syl2anc |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) ∧ ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐴 ∧ ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑦 𝐵 ) ∈ Comp ) ) → ( ∪ 𝑥 ∈ 𝑦 𝐵 ∪ ∪ 𝑥 ∈ { 𝑧 } 𝐵 ) ∈ V ) |
79 |
40 78
|
eqeltrid |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) ∧ ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐴 ∧ ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑦 𝐵 ) ∈ Comp ) ) → ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ∈ V ) |
80 |
|
resttop |
⊢ ( ( 𝐽 ∈ Top ∧ ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ∈ V ) → ( 𝐽 ↾t ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ) ∈ Top ) |
81 |
39 79 80
|
syl2anc |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) ∧ ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐴 ∧ ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑦 𝐵 ) ∈ Comp ) ) → ( 𝐽 ↾t ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ) ∈ Top ) |
82 |
|
eqid |
⊢ ∪ 𝐽 = ∪ 𝐽 |
83 |
82
|
restin |
⊢ ( ( 𝐽 ∈ Top ∧ ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ∈ V ) → ( 𝐽 ↾t ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ) = ( 𝐽 ↾t ( ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ∩ ∪ 𝐽 ) ) ) |
84 |
39 79 83
|
syl2anc |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) ∧ ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐴 ∧ ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑦 𝐵 ) ∈ Comp ) ) → ( 𝐽 ↾t ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ) = ( 𝐽 ↾t ( ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ∩ ∪ 𝐽 ) ) ) |
85 |
84
|
unieqd |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) ∧ ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐴 ∧ ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑦 𝐵 ) ∈ Comp ) ) → ∪ ( 𝐽 ↾t ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ) = ∪ ( 𝐽 ↾t ( ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ∩ ∪ 𝐽 ) ) ) |
86 |
|
inss2 |
⊢ ( ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ∩ ∪ 𝐽 ) ⊆ ∪ 𝐽 |
87 |
86 1
|
sseqtrri |
⊢ ( ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ∩ ∪ 𝐽 ) ⊆ 𝑋 |
88 |
1
|
restuni |
⊢ ( ( 𝐽 ∈ Top ∧ ( ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ∩ ∪ 𝐽 ) ⊆ 𝑋 ) → ( ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ∩ ∪ 𝐽 ) = ∪ ( 𝐽 ↾t ( ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ∩ ∪ 𝐽 ) ) ) |
89 |
39 87 88
|
sylancl |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) ∧ ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐴 ∧ ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑦 𝐵 ) ∈ Comp ) ) → ( ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ∩ ∪ 𝐽 ) = ∪ ( 𝐽 ↾t ( ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ∩ ∪ 𝐽 ) ) ) |
90 |
85 89
|
eqtr4d |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) ∧ ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐴 ∧ ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑦 𝐵 ) ∈ Comp ) ) → ∪ ( 𝐽 ↾t ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ) = ( ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ∩ ∪ 𝐽 ) ) |
91 |
53
|
uneq2i |
⊢ ( ∪ 𝑥 ∈ 𝑦 𝐵 ∪ ∪ 𝑥 ∈ { 𝑧 } 𝐵 ) = ( ∪ 𝑥 ∈ 𝑦 𝐵 ∪ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) |
92 |
40 91
|
eqtri |
⊢ ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 = ( ∪ 𝑥 ∈ 𝑦 𝐵 ∪ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) |
93 |
92
|
ineq1i |
⊢ ( ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ∩ ∪ 𝐽 ) = ( ( ∪ 𝑥 ∈ 𝑦 𝐵 ∪ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ∩ ∪ 𝐽 ) |
94 |
|
indir |
⊢ ( ( ∪ 𝑥 ∈ 𝑦 𝐵 ∪ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ∩ ∪ 𝐽 ) = ( ( ∪ 𝑥 ∈ 𝑦 𝐵 ∩ ∪ 𝐽 ) ∪ ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ ∪ 𝐽 ) ) |
95 |
93 94
|
eqtri |
⊢ ( ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ∩ ∪ 𝐽 ) = ( ( ∪ 𝑥 ∈ 𝑦 𝐵 ∩ ∪ 𝐽 ) ∪ ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ ∪ 𝐽 ) ) |
96 |
90 95
|
eqtrdi |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) ∧ ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐴 ∧ ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑦 𝐵 ) ∈ Comp ) ) → ∪ ( 𝐽 ↾t ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ) = ( ( ∪ 𝑥 ∈ 𝑦 𝐵 ∩ ∪ 𝐽 ) ∪ ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ ∪ 𝐽 ) ) ) |
97 |
|
inss1 |
⊢ ( ∪ 𝑥 ∈ 𝑦 𝐵 ∩ ∪ 𝐽 ) ⊆ ∪ 𝑥 ∈ 𝑦 𝐵 |
98 |
|
ssun1 |
⊢ ∪ 𝑥 ∈ 𝑦 𝐵 ⊆ ( ∪ 𝑥 ∈ 𝑦 𝐵 ∪ ∪ 𝑥 ∈ { 𝑧 } 𝐵 ) |
99 |
98 40
|
sseqtrri |
⊢ ∪ 𝑥 ∈ 𝑦 𝐵 ⊆ ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 |
100 |
97 99
|
sstri |
⊢ ( ∪ 𝑥 ∈ 𝑦 𝐵 ∩ ∪ 𝐽 ) ⊆ ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 |
101 |
100
|
a1i |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) ∧ ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐴 ∧ ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑦 𝐵 ) ∈ Comp ) ) → ( ∪ 𝑥 ∈ 𝑦 𝐵 ∩ ∪ 𝐽 ) ⊆ ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ) |
102 |
|
restabs |
⊢ ( ( 𝐽 ∈ Top ∧ ( ∪ 𝑥 ∈ 𝑦 𝐵 ∩ ∪ 𝐽 ) ⊆ ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ∧ ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ∈ V ) → ( ( 𝐽 ↾t ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ) ↾t ( ∪ 𝑥 ∈ 𝑦 𝐵 ∩ ∪ 𝐽 ) ) = ( 𝐽 ↾t ( ∪ 𝑥 ∈ 𝑦 𝐵 ∩ ∪ 𝐽 ) ) ) |
103 |
39 101 79 102
|
syl3anc |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) ∧ ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐴 ∧ ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑦 𝐵 ) ∈ Comp ) ) → ( ( 𝐽 ↾t ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ) ↾t ( ∪ 𝑥 ∈ 𝑦 𝐵 ∩ ∪ 𝐽 ) ) = ( 𝐽 ↾t ( ∪ 𝑥 ∈ 𝑦 𝐵 ∩ ∪ 𝐽 ) ) ) |
104 |
82
|
restin |
⊢ ( ( 𝐽 ∈ Top ∧ ∪ 𝑥 ∈ 𝑦 𝐵 ∈ V ) → ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑦 𝐵 ) = ( 𝐽 ↾t ( ∪ 𝑥 ∈ 𝑦 𝐵 ∩ ∪ 𝐽 ) ) ) |
105 |
39 45 104
|
syl2anc |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) ∧ ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐴 ∧ ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑦 𝐵 ) ∈ Comp ) ) → ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑦 𝐵 ) = ( 𝐽 ↾t ( ∪ 𝑥 ∈ 𝑦 𝐵 ∩ ∪ 𝐽 ) ) ) |
106 |
103 105
|
eqtr4d |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) ∧ ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐴 ∧ ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑦 𝐵 ) ∈ Comp ) ) → ( ( 𝐽 ↾t ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ) ↾t ( ∪ 𝑥 ∈ 𝑦 𝐵 ∩ ∪ 𝐽 ) ) = ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑦 𝐵 ) ) |
107 |
106 41
|
eqeltrd |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) ∧ ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐴 ∧ ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑦 𝐵 ) ∈ Comp ) ) → ( ( 𝐽 ↾t ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ) ↾t ( ∪ 𝑥 ∈ 𝑦 𝐵 ∩ ∪ 𝐽 ) ) ∈ Comp ) |
108 |
|
inss1 |
⊢ ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ ∪ 𝐽 ) ⊆ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 |
109 |
|
ssun2 |
⊢ ∪ 𝑥 ∈ { 𝑧 } 𝐵 ⊆ ( ∪ 𝑥 ∈ 𝑦 𝐵 ∪ ∪ 𝑥 ∈ { 𝑧 } 𝐵 ) |
110 |
109 40
|
sseqtrri |
⊢ ∪ 𝑥 ∈ { 𝑧 } 𝐵 ⊆ ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 |
111 |
53 110
|
eqsstrri |
⊢ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ⊆ ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 |
112 |
108 111
|
sstri |
⊢ ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ ∪ 𝐽 ) ⊆ ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 |
113 |
112
|
a1i |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) ∧ ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐴 ∧ ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑦 𝐵 ) ∈ Comp ) ) → ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ ∪ 𝐽 ) ⊆ ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ) |
114 |
|
restabs |
⊢ ( ( 𝐽 ∈ Top ∧ ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ ∪ 𝐽 ) ⊆ ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ∧ ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ∈ V ) → ( ( 𝐽 ↾t ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ) ↾t ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ ∪ 𝐽 ) ) = ( 𝐽 ↾t ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ ∪ 𝐽 ) ) ) |
115 |
39 113 79 114
|
syl3anc |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) ∧ ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐴 ∧ ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑦 𝐵 ) ∈ Comp ) ) → ( ( 𝐽 ↾t ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ) ↾t ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ ∪ 𝐽 ) ) = ( 𝐽 ↾t ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ ∪ 𝐽 ) ) ) |
116 |
82
|
restin |
⊢ ( ( 𝐽 ∈ Top ∧ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∈ V ) → ( 𝐽 ↾t ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) = ( 𝐽 ↾t ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ ∪ 𝐽 ) ) ) |
117 |
39 75 116
|
syl2anc |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) ∧ ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐴 ∧ ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑦 𝐵 ) ∈ Comp ) ) → ( 𝐽 ↾t ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) = ( 𝐽 ↾t ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ ∪ 𝐽 ) ) ) |
118 |
115 117
|
eqtr4d |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) ∧ ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐴 ∧ ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑦 𝐵 ) ∈ Comp ) ) → ( ( 𝐽 ↾t ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ) ↾t ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ ∪ 𝐽 ) ) = ( 𝐽 ↾t ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ) |
119 |
118 71
|
eqeltrd |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) ∧ ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐴 ∧ ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑦 𝐵 ) ∈ Comp ) ) → ( ( 𝐽 ↾t ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ) ↾t ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ ∪ 𝐽 ) ) ∈ Comp ) |
120 |
|
eqid |
⊢ ∪ ( 𝐽 ↾t ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ) = ∪ ( 𝐽 ↾t ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ) |
121 |
120
|
uncmp |
⊢ ( ( ( ( 𝐽 ↾t ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ) ∈ Top ∧ ∪ ( 𝐽 ↾t ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ) = ( ( ∪ 𝑥 ∈ 𝑦 𝐵 ∩ ∪ 𝐽 ) ∪ ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ ∪ 𝐽 ) ) ) ∧ ( ( ( 𝐽 ↾t ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ) ↾t ( ∪ 𝑥 ∈ 𝑦 𝐵 ∩ ∪ 𝐽 ) ) ∈ Comp ∧ ( ( 𝐽 ↾t ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ) ↾t ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ ∪ 𝐽 ) ) ∈ Comp ) ) → ( 𝐽 ↾t ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ) ∈ Comp ) |
122 |
81 96 107 119 121
|
syl22anc |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) ∧ ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐴 ∧ ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑦 𝐵 ) ∈ Comp ) ) → ( 𝐽 ↾t ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ) ∈ Comp ) |
123 |
122
|
exp32 |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) → ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐴 → ( ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑦 𝐵 ) ∈ Comp → ( 𝐽 ↾t ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ) ∈ Comp ) ) ) |
124 |
123
|
a2d |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) → ( ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐴 → ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑦 𝐵 ) ∈ Comp ) → ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐴 → ( 𝐽 ↾t ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ) ∈ Comp ) ) ) |
125 |
38 124
|
syl5 |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) → ( ( 𝑦 ⊆ 𝐴 → ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑦 𝐵 ) ∈ Comp ) → ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐴 → ( 𝐽 ↾t ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ) ∈ Comp ) ) ) |
126 |
125
|
a2i |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) → ( 𝑦 ⊆ 𝐴 → ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑦 𝐵 ) ∈ Comp ) ) → ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) → ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐴 → ( 𝐽 ↾t ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ) ∈ Comp ) ) ) |
127 |
126
|
a1i |
⊢ ( 𝑦 ∈ Fin → ( ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) → ( 𝑦 ⊆ 𝐴 → ( 𝐽 ↾t ∪ 𝑥 ∈ 𝑦 𝐵 ) ∈ Comp ) ) → ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) → ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐴 → ( 𝐽 ↾t ∪ 𝑥 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ) ∈ Comp ) ) ) ) |
128 |
11 17 23 29 34 127
|
findcard2 |
⊢ ( 𝐴 ∈ Fin → ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) → ( 𝐴 ⊆ 𝐴 → ( 𝐽 ↾t ∪ 𝑥 ∈ 𝐴 𝐵 ) ∈ Comp ) ) ) |
129 |
3 128
|
mpcom |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) → ( 𝐴 ⊆ 𝐴 → ( 𝐽 ↾t ∪ 𝑥 ∈ 𝐴 𝐵 ) ∈ Comp ) ) |
130 |
2 129
|
mpi |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐽 ↾t 𝐵 ) ∈ Comp ) → ( 𝐽 ↾t ∪ 𝑥 ∈ 𝐴 𝐵 ) ∈ Comp ) |